# Maple integration test file: "4 Trig functions\4.2 Cosine\4.2.1.2 (g sin)^p (a+b cos)^m.txt"

lst:=[

# Integrands of the form (g Sin[e+f x])^p (a+b Cos[e+f x])^m

# Integrands of the form (g Sin[e+f x])^p (a+b Cos[e+f x])^m when a^2-b^2=0

# Integrands of the form Sin[e+f x]^p (a+a Cos[e+f x])^m
[sin(x)^4/(a+a*cos(x)),x,3,1/2*x/a-1/2*cos(x)*sin(x)/a-1/3*sin(x)^3/a],
[sin(x)^3/(a+a*cos(x)),x,2,-cos(x)/a+1/2*cos(x)^2/a],
[sin(x)^2/(a+a*cos(x)),x,2,x/a-sin(x)/a],
[sin(x)/(a+a*cos(x)),x,2,-log(1+cos(x))/a],
[1/(a+a*cos(x)),x,1,sin(x)/(a+a*cos(x))],
[csc(x)/(a+a*cos(x)),x,4,-1/2*arctanh(cos(x))/a+1/2/(a+a*cos(x))],
[csc(x)^2/(a+a*cos(x)),x,3,-2/3*cot(x)/a+1/3*csc(x)/(a+a*cos(x))],
[csc(x)^3/(a+a*cos(x)),x,4,-3/8*arctanh(cos(x))/a+(-1/8)/(a-a*cos(x))+1/8*a/(a+a*cos(x))^2+1/4/(a+a*cos(x))],
[csc(x)^4/(a+a*cos(x)),x,3,-4/5*cot(x)/a-4/15*cot(x)^3/a+1/5*csc(x)^3/(a+a*cos(x))],
[sin(2*x)/(1+cos(2*x)),x,2,-log(cos(x)),-1/2*log(1+cos(2*x))],
[sin(2*x)/(1-cos(2*x)),x,2,log(sin(x)),1/2*log(1-cos(2*x))],
[sin(x)/(1+cos(x))^2,x,2,1/(1+cos(x))],
[sin(x)/(1-cos(x))^2,x,2,(-1)/(1-cos(x))],
[sin(x)^2/(1+cos(x))^2,x,2,-x+2*sin(x)/(1+cos(x))],
[sin(x)^2/(1-cos(x))^2,x,2,-x-2*sin(x)/(1-cos(x))],
[sin(x)^3/(1+cos(x))^2,x,3,cos(x)-2*log(1+cos(x))],
[sin(x)^3/(1-cos(x))^2,x,3,cos(x)+2*log(1-cos(x))],
[sin(x)/(1+cos(x))^3,x,2,1/2/(1+cos(x))^2],
[sin(x)/(1-cos(x))^3,x,2,(-1/2)/(1-cos(x))^2],
[sin(x)^2/(1+cos(x))^3,x,1,1/3*sin(x)^3/(1+cos(x))^3],
[sin(x)^2/(1-cos(x))^3,x,1,-1/3*sin(x)^3/(1-cos(x))^3],
[sin(x)^3/(1+cos(x))^3,x,3,2/(1+cos(x))+log(1+cos(x))],
[sin(x)^3/(1-cos(x))^3,x,3,(-2)/(1-cos(x))-log(1-cos(x))],

# Integrands of the form Sin[e+f x]^p (a+a Cos[e+f x])^(m/2)

# Integrands of the form (g Sin[e+f x])^(p/2) (a+a Cos[e+f x])^m

# Integrands of the form (g Sin[e+f x])^(p/2) (a+a Cos[e+f x])^(m/2)

# Integrands of the form (g Sin[e+f x])^p (a+b Cos[e+f x])^m

# Integrands of the form Sin[e+f x]^p (a+b Cos[e+f x])^m

# n>0

# n<0
[sin(x)^4/(a+b*cos(x)),x,5,-1/2*a*(2*a^2-3*b^2)*x/b^4+2*(a-b)^(3/2)*(a+b)^(3/2)*arctan(sqrt(a-b)*tan(1/2*x)/sqrt(a+b))/b^4+1/2*(2*(a^2-b^2)-a*b*cos(x))*sin(x)/b^3-1/3*sin(x)^3/b],
[sin(x)^3/(a+b*cos(x)),x,3,-a*cos(x)/b^2+1/2*cos(x)^2/b+(a^2-b^2)*log(a+b*cos(x))/b^3],
[sin(x)^2/(a+b*cos(x)),x,4,a*x/b^2-sin(x)/b-2*arctan(sqrt(a-b)*tan(1/2*x)/sqrt(a+b))*sqrt(a-b)*sqrt(a+b)/b^2],
[sin(x)/(a+b*cos(x)),x,2,-log(a+b*cos(x))/b],
[1/(a+b*cos(x)),x,2,2*arctan(sqrt(a-b)*tan(1/2*x)/sqrt(a+b))/(sqrt(a-b)*sqrt(a+b))],
[csc(x)/(a+b*cos(x)),x,6,1/2*log(1-cos(x))/(a+b)-1/2*log(1+cos(x))/(a-b)+b*log(a+b*cos(x))/(a^2-b^2)],
[csc(x)^2/(a+b*cos(x)),x,4,-2*b^2*arctan(sqrt(a-b)*tan(1/2*x)/sqrt(a+b))/((a-b)^(3/2)*(a+b)^(3/2))+(b-a*cos(x))*csc(x)/(a^2-b^2)],
[csc(x)^3/(a+b*cos(x)),x,4,1/2*(b-a*cos(x))*csc(x)^2/(a^2-b^2)+1/4*(a+2*b)*log(1-cos(x))/(a+b)^2-1/4*(a-2*b)*log(1+cos(x))/(a-b)^2-b^3*log(a+b*cos(x))/(a^2-b^2)^2],
[csc(x)^4/(a+b*cos(x)),x,5,2*b^4*arctan(sqrt(a-b)*tan(1/2*x)/sqrt(a+b))/((a-b)^(5/2)*(a+b)^(5/2))-1/3*(3*b^3+a*(2*a^2-5*b^2)*cos(x))*csc(x)/(a^2-b^2)^2+1/3*(b-a*cos(x))*csc(x)^3/(a^2-b^2)],

# Integrands of the form Sin[e+f x]^p (a+b Cos[e+f x])^(m/2)

# Integrands of the form (g Sin[e+f x])^(p/2) (a+b Cos[e+f x])^m

# m>0
[(a+b*cos(c+d*x))*(e*sin(c+d*x))^(7/2),x,5,-2/7*a*e*cos(c+d*x)*(e*sin(c+d*x))^(5/2)/d+2/9*b*(e*sin(c+d*x))^(9/2)/(d*e)+10/21*a*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*sqrt(e*sin(c+d*x)))-10/21*a*e^3*cos(c+d*x)*sqrt(e*sin(c+d*x))/d],
[(a+b*cos(c+d*x))*(e*sin(c+d*x))^(5/2),x,4,-2/5*a*e*cos(c+d*x)*(e*sin(c+d*x))^(3/2)/d+2/7*b*(e*sin(c+d*x))^(7/2)/(d*e)+6/5*a*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*sqrt(sin(c+d*x)))],
[(a+b*cos(c+d*x))*(e*sin(c+d*x))^(3/2),x,4,2/5*b*(e*sin(c+d*x))^(5/2)/(d*e)+2/3*a*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*sqrt(e*sin(c+d*x)))-2/3*a*e*cos(c+d*x)*sqrt(e*sin(c+d*x))/d],
[(a+b*cos(c+d*x))*(e*sin(c+d*x))^(1/2),x,3,2/3*b*(e*sin(c+d*x))^(3/2)/(d*e)+2*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*sqrt(sin(c+d*x)))],
[(a+b*cos(c+d*x))/(e*sin(c+d*x))^(1/2),x,3,2*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*sqrt(e*sin(c+d*x)))+2*b*sqrt(e*sin(c+d*x))/(d*e)],
[(a+b*cos(c+d*x))/(e*sin(c+d*x))^(3/2),x,4,-2*b/(d*e*sqrt(e*sin(c+d*x)))-2*a*cos(c+d*x)/(d*e*sqrt(e*sin(c+d*x)))-2*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*e^2*sqrt(sin(c+d*x)))],
[(a+b*cos(c+d*x))/(e*sin(c+d*x))^(5/2),x,4,-2/3*b/(d*e*(e*sin(c+d*x))^(3/2))-2/3*a*cos(c+d*x)/(d*e*(e*sin(c+d*x))^(3/2))+2/3*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*e^2*sqrt(e*sin(c+d*x)))],
[(a+b*cos(c+d*x))/(e*sin(c+d*x))^(7/2),x,5,-2/5*b/(d*e*(e*sin(c+d*x))^(5/2))-2/5*a*cos(c+d*x)/(d*e*(e*sin(c+d*x))^(5/2))-6/5*a*cos(c+d*x)/(d*e^3*sqrt(e*sin(c+d*x)))-6/5*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*e^4*sqrt(sin(c+d*x)))],
[(a+b*cos(c+d*x))^2*(e*sin(c+d*x))^(7/2),x,6,-2/77*(11*a^2+2*b^2)*e*cos(c+d*x)*(e*sin(c+d*x))^(5/2)/d+26/99*a*b*(e*sin(c+d*x))^(9/2)/(d*e)+2/11*b*(a+b*cos(c+d*x))*(e*sin(c+d*x))^(9/2)/(d*e)+10/231*(11*a^2+2*b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*sqrt(e*sin(c+d*x)))-10/231*(11*a^2+2*b^2)*e^3*cos(c+d*x)*sqrt(e*sin(c+d*x))/d],
[(a+b*cos(c+d*x))^2*(e*sin(c+d*x))^(5/2),x,5,-2/45*(9*a^2+2*b^2)*e*cos(c+d*x)*(e*sin(c+d*x))^(3/2)/d+22/63*a*b*(e*sin(c+d*x))^(7/2)/(d*e)+2/9*b*(a+b*cos(c+d*x))*(e*sin(c+d*x))^(7/2)/(d*e)+2/15*(9*a^2+2*b^2)*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*sqrt(sin(c+d*x)))],
[(a+b*cos(c+d*x))^2*(e*sin(c+d*x))^(3/2),x,5,18/35*a*b*(e*sin(c+d*x))^(5/2)/(d*e)+2/7*b*(a+b*cos(c+d*x))*(e*sin(c+d*x))^(5/2)/(d*e)+2/21*(7*a^2+2*b^2)*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*sqrt(e*sin(c+d*x)))-2/21*(7*a^2+2*b^2)*e*cos(c+d*x)*sqrt(e*sin(c+d*x))/d],
[(a+b*cos(c+d*x))^2*(e*sin(c+d*x))^(1/2),x,4,14/15*a*b*(e*sin(c+d*x))^(3/2)/(d*e)+2/5*b*(a+b*cos(c+d*x))*(e*sin(c+d*x))^(3/2)/(d*e)+2/5*(5*a^2+2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*sqrt(sin(c+d*x)))],
[(a+b*cos(c+d*x))^2/(e*sin(c+d*x))^(1/2),x,4,2/3*(3*a^2+2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*sqrt(e*sin(c+d*x)))+10/3*a*b*sqrt(e*sin(c+d*x))/(d*e)+2/3*b*(a+b*cos(c+d*x))*sqrt(e*sin(c+d*x))/(d*e)],
[(a+b*cos(c+d*x))^2/(e*sin(c+d*x))^(3/2),x,4,-2*a*b*(e*sin(c+d*x))^(3/2)/(d*e^3)-2*(b+a*cos(c+d*x))*(a+b*cos(c+d*x))/(d*e*sqrt(e*sin(c+d*x)))-2*(a^2+2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*e^2*sqrt(sin(c+d*x)))],
[(a+b*cos(c+d*x))^2/(e*sin(c+d*x))^(5/2),x,4,-2/3*(b+a*cos(c+d*x))*(a+b*cos(c+d*x))/(d*e*(e*sin(c+d*x))^(3/2))+2/3*(a^2-2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*e^2*sqrt(e*sin(c+d*x)))-2/3*a*b*sqrt(e*sin(c+d*x))/(d*e^3)],
[(a+b*cos(c+d*x))^2/(e*sin(c+d*x))^(7/2),x,5,-2/5*(b+a*cos(c+d*x))*(a+b*cos(c+d*x))/(d*e*(e*sin(c+d*x))^(5/2))-2/5*a*b/(d*e^3*sqrt(e*sin(c+d*x)))-2/5*(3*a^2-2*b^2)*cos(c+d*x)/(d*e^3*sqrt(e*sin(c+d*x)))-2/5*(3*a^2-2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*e^4*sqrt(sin(c+d*x)))],
[(a+b*cos(c+d*x))^3*(e*sin(c+d*x))^(7/2),x,7,-2/77*a*(11*a^2+6*b^2)*e*cos(c+d*x)*(e*sin(c+d*x))^(5/2)/d+2/1287*b*(177*a^2+44*b^2)*(e*sin(c+d*x))^(9/2)/(d*e)+34/143*a*b*(a+b*cos(c+d*x))*(e*sin(c+d*x))^(9/2)/(d*e)+2/13*b*(a+b*cos(c+d*x))^2*(e*sin(c+d*x))^(9/2)/(d*e)+10/231*a*(11*a^2+6*b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*sqrt(e*sin(c+d*x)))-10/231*a*(11*a^2+6*b^2)*e^3*cos(c+d*x)*sqrt(e*sin(c+d*x))/d],
[(a+b*cos(c+d*x))^3*(e*sin(c+d*x))^(5/2),x,6,-2/15*a*(3*a^2+2*b^2)*e*cos(c+d*x)*(e*sin(c+d*x))^(3/2)/d+2/231*b*(43*a^2+12*b^2)*(e*sin(c+d*x))^(7/2)/(d*e)+10/33*a*b*(a+b*cos(c+d*x))*(e*sin(c+d*x))^(7/2)/(d*e)+2/11*b*(a+b*cos(c+d*x))^2*(e*sin(c+d*x))^(7/2)/(d*e)+2/5*a*(3*a^2+2*b^2)*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*sqrt(sin(c+d*x)))],
[(a+b*cos(c+d*x))^3*(e*sin(c+d*x))^(3/2),x,6,2/315*b*(89*a^2+28*b^2)*(e*sin(c+d*x))^(5/2)/(d*e)+26/63*a*b*(a+b*cos(c+d*x))*(e*sin(c+d*x))^(5/2)/(d*e)+2/9*b*(a+b*cos(c+d*x))^2*(e*sin(c+d*x))^(5/2)/(d*e)+2/21*a*(7*a^2+6*b^2)*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*sqrt(e*sin(c+d*x)))-2/21*a*(7*a^2+6*b^2)*e*cos(c+d*x)*sqrt(e*sin(c+d*x))/d],
[(a+b*cos(c+d*x))^3*(e*sin(c+d*x))^(1/2),x,5,2/105*b*(57*a^2+20*b^2)*(e*sin(c+d*x))^(3/2)/(d*e)+22/35*a*b*(a+b*cos(c+d*x))*(e*sin(c+d*x))^(3/2)/(d*e)+2/7*b*(a+b*cos(c+d*x))^2*(e*sin(c+d*x))^(3/2)/(d*e)+2/5*a*(5*a^2+6*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*sqrt(sin(c+d*x)))],
[(a+b*cos(c+d*x))^3/(e*sin(c+d*x))^(1/2),x,5,2*a*(a^2+2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*sqrt(e*sin(c+d*x)))+2/5*b*(11*a^2+4*b^2)*sqrt(e*sin(c+d*x))/(d*e)+6/5*a*b*(a+b*cos(c+d*x))*sqrt(e*sin(c+d*x))/(d*e)+2/5*b*(a+b*cos(c+d*x))^2*sqrt(e*sin(c+d*x))/(d*e)],
[(a+b*cos(c+d*x))^3/(e*sin(c+d*x))^(3/2),x,5,-2/3*b*(3*a^2+4*b^2)*(e*sin(c+d*x))^(3/2)/(d*e^3)-2*a*b*(a+b*cos(c+d*x))*(e*sin(c+d*x))^(3/2)/(d*e^3)-2*(b+a*cos(c+d*x))*(a+b*cos(c+d*x))^2/(d*e*sqrt(e*sin(c+d*x)))-2*a*(a^2+6*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*e^2*sqrt(sin(c+d*x)))],
[(a+b*cos(c+d*x))^3/(e*sin(c+d*x))^(5/2),x,5,-2/3*(b+a*cos(c+d*x))*(a+b*cos(c+d*x))^2/(d*e*(e*sin(c+d*x))^(3/2))+2/3*a*(a^2-6*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*e^2*sqrt(e*sin(c+d*x)))-2/3*b*(a^2+4*b^2)*sqrt(e*sin(c+d*x))/(d*e^3)-2/3*a*b*(a+b*cos(c+d*x))*sqrt(e*sin(c+d*x))/(d*e^3)],
[(a+b*cos(c+d*x))^3/(e*sin(c+d*x))^(7/2),x,5,-2/5*(b+a*cos(c+d*x))*(a+b*cos(c+d*x))^2/(d*e*(e*sin(c+d*x))^(5/2))-2/5*b*(3*a^2-4*b^2)*(e*sin(c+d*x))^(3/2)/(d*e^5)+2/5*(a+b*cos(c+d*x))*(a*b-(3*a^2-4*b^2)*cos(c+d*x))/(d*e^3*sqrt(e*sin(c+d*x)))-6/5*a*(a^2-2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(d*e^4*sqrt(sin(c+d*x)))],
[(a+b*cos(c+d*x))^3/(e*sin(c+d*x))^(9/2),x,5,-2/7*(b+a*cos(c+d*x))*(a+b*cos(c+d*x))^2/(d*e*(e*sin(c+d*x))^(7/2))-2/21*(a+b*cos(c+d*x))*(a*b+(5*a^2-4*b^2)*cos(c+d*x))/(d*e^3*(e*sin(c+d*x))^(3/2))+2/21*a*(5*a^2-6*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(d*e^4*sqrt(e*sin(c+d*x)))-2/21*b*(5*a^2-4*b^2)*sqrt(e*sin(c+d*x))/(d*e^5)],

# m<0
[(e*sin(c+d*x))^(11/2)/(a+b*cos(c+d*x)),x,15,(-a^2+b^2)^(9/4)*e^(11/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(11/2)*d)+(-a^2+b^2)^(9/4)*e^(11/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(11/2)*d)+2/35*e^3*(7*(a^2-b^2)-5*a*b*cos(c+d*x))*(e*sin(c+d*x))^(5/2)/(b^3*d)-2/9*e*(e*sin(c+d*x))^(9/2)/(b*d)+2/21*a*(21*a^4-49*a^2*b^2+33*b^4)*e^6*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(b^6*d*sqrt(e*sin(c+d*x)))-a*(a^2-b^2)^3*e^6*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^6*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))-a*(a^2-b^2)^3*e^6*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^6*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))-2/21*e^5*(21*(a^2-b^2)^2-a*b*(7*a^2-12*b^2)*cos(c+d*x))*sqrt(e*sin(c+d*x))/(b^5*d)],
[(e*sin(c+d*x))^(9/2)/(a+b*cos(c+d*x)),x,14,-(-a^2+b^2)^(7/4)*e^(9/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(9/2)*d)+(-a^2+b^2)^(7/4)*e^(9/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(9/2)*d)+2/15*e^3*(5*(a^2-b^2)-3*a*b*cos(c+d*x))*(e*sin(c+d*x))^(3/2)/(b^3*d)-2/7*e*(e*sin(c+d*x))^(7/2)/(b*d)+a*(a^2-b^2)^2*e^5*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^5*d*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))+a*(a^2-b^2)^2*e^5*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^5*d*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-2/5*a*(5*a^2-8*b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(b^4*d*sqrt(sin(c+d*x)))],
[(e*sin(c+d*x))^(7/2)/(a+b*cos(c+d*x)),x,14,(-a^2+b^2)^(5/4)*e^(7/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(7/2)*d)+(-a^2+b^2)^(5/4)*e^(7/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(7/2)*d)-2/5*e*(e*sin(c+d*x))^(5/2)/(b*d)-2/3*a*(3*a^2-4*b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(b^4*d*sqrt(e*sin(c+d*x)))+a*(a^2-b^2)^2*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^4*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))+a*(a^2-b^2)^2*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^4*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))+2/3*e^3*(3*(a^2-b^2)-a*b*cos(c+d*x))*sqrt(e*sin(c+d*x))/(b^3*d)],
[(e*sin(c+d*x))^(5/2)/(a+b*cos(c+d*x)),x,13,-(-a^2+b^2)^(3/4)*e^(5/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(5/2)*d)+(-a^2+b^2)^(3/4)*e^(5/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(5/2)*d)-2/3*e*(e*sin(c+d*x))^(3/2)/(b*d)-a*(a^2-b^2)*e^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^3*d*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-a*(a^2-b^2)*e^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^3*d*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))+2*a*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(b^2*d*sqrt(sin(c+d*x)))],
[(e*sin(c+d*x))^(3/2)/(a+b*cos(c+d*x)),x,13,(-a^2+b^2)^(1/4)*e^(3/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(3/2)*d)+(-a^2+b^2)^(1/4)*e^(3/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(3/2)*d)+2*a*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(b^2*d*sqrt(e*sin(c+d*x)))-a*(a^2-b^2)*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^2*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))-a*(a^2-b^2)*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^2*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))-2*e*sqrt(e*sin(c+d*x))/(b*d)],
[(e*sin(c+d*x))^(1/2)/(a+b*cos(c+d*x)),x,9,-arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(e)/((-a^2+b^2)^(1/4)*d*sqrt(b))+arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(e)/((-a^2+b^2)^(1/4)*d*sqrt(b))+a*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b*d*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))+a*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b*d*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))],
[1/((a+b*cos(c+d*x))*(e*sin(c+d*x))^(1/2)),x,9,arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(b)/((-a^2+b^2)^(3/4)*d*sqrt(e))+arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(b)/((-a^2+b^2)^(3/4)*d*sqrt(e))+a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))+a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))],
[1/((a+b*cos(c+d*x))*(e*sin(c+d*x))^(3/2)),x,13,-b^(3/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(5/4)*d*e^(3/2))+b^(3/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(5/4)*d*e^(3/2))+2*(b-a*cos(c+d*x))/((a^2-b^2)*d*e*sqrt(e*sin(c+d*x)))-a*b*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)*d*e*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-a*b*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)*d*e*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-2*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/((a^2-b^2)*d*e^2*sqrt(sin(c+d*x)))],
[1/((a+b*cos(c+d*x))*(e*sin(c+d*x))^(5/2)),x,13,b^(5/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(7/4)*d*e^(5/2))+b^(5/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(7/4)*d*e^(5/2))+2/3*(b-a*cos(c+d*x))/((a^2-b^2)*d*e*(e*sin(c+d*x))^(3/2))+2/3*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)*d*e^2*sqrt(e*sin(c+d*x)))-a*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)*d*e^2*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))-a*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)*d*e^2*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))],
[1/((a+b*cos(c+d*x))*(e*sin(c+d*x))^(7/2)),x,14,-b^(7/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(9/4)*d*e^(7/2))+b^(7/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(9/4)*d*e^(7/2))+2/5*(b-a*cos(c+d*x))/((a^2-b^2)*d*e*(e*sin(c+d*x))^(5/2))-2/5*(5*b^3+a*(3*a^2-8*b^2)*cos(c+d*x))/((a^2-b^2)^2*d*e^3*sqrt(e*sin(c+d*x)))+a*b^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^2*d*e^3*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))+a*b^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^2*d*e^3*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-2/5*a*(3*a^2-8*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/((a^2-b^2)^2*d*e^4*sqrt(sin(c+d*x)))],
[(e*sin(c+d*x))^(11/2)/(a+b*cos(c+d*x))^2,x,15,9/2*a*(-a^2+b^2)^(5/4)*e^(11/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(11/2)*d)+9/2*a*(-a^2+b^2)^(5/4)*e^(11/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(11/2)*d)-9/35*e^3*(7*a-5*b*cos(c+d*x))*(e*sin(c+d*x))^(5/2)/(b^3*d)+e*(e*sin(c+d*x))^(9/2)/(b*d*(a+b*cos(c+d*x)))-3/7*(21*a^4-28*a^2*b^2+5*b^4)*e^6*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(b^6*d*sqrt(e*sin(c+d*x)))+9/2*a^2*(a^2-b^2)^2*e^6*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^6*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))+9/2*a^2*(a^2-b^2)^2*e^6*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^6*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))+3/7*e^5*(21*a*(a^2-b^2)-b*(7*a^2-5*b^2)*cos(c+d*x))*sqrt(e*sin(c+d*x))/(b^5*d)],
[(e*sin(c+d*x))^(9/2)/(a+b*cos(c+d*x))^2,x,14,-7/2*a*(-a^2+b^2)^(3/4)*e^(9/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(9/2)*d)+7/2*a*(-a^2+b^2)^(3/4)*e^(9/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(9/2)*d)-7/15*e^3*(5*a-3*b*cos(c+d*x))*(e*sin(c+d*x))^(3/2)/(b^3*d)+e*(e*sin(c+d*x))^(7/2)/(b*d*(a+b*cos(c+d*x)))-7/2*a^2*(a^2-b^2)*e^5*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^5*d*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-7/2*a^2*(a^2-b^2)*e^5*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^5*d*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))+7/5*(5*a^2-3*b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(b^4*d*sqrt(sin(c+d*x)))],
[(e*sin(c+d*x))^(7/2)/(a+b*cos(c+d*x))^2,x,14,5/2*a*(-a^2+b^2)^(1/4)*e^(7/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(7/2)*d)+5/2*a*(-a^2+b^2)^(1/4)*e^(7/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(7/2)*d)+e*(e*sin(c+d*x))^(5/2)/(b*d*(a+b*cos(c+d*x)))+5/3*(3*a^2-b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(b^4*d*sqrt(e*sin(c+d*x)))-5/2*a^2*(a^2-b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^4*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))-5/2*a^2*(a^2-b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^4*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))-5/3*e^3*(3*a-b*cos(c+d*x))*sqrt(e*sin(c+d*x))/(b^3*d)],
[(e*sin(c+d*x))^(5/2)/(a+b*cos(c+d*x))^2,x,13,-3/2*a*e^(5/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(5/2)*(-a^2+b^2)^(1/4)*d)+3/2*a*e^(5/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(5/2)*(-a^2+b^2)^(1/4)*d)+e*(e*sin(c+d*x))^(3/2)/(b*d*(a+b*cos(c+d*x)))+3/2*a^2*e^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^3*d*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))+3/2*a^2*e^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^3*d*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-3*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(b^2*d*sqrt(sin(c+d*x)))],
[(e*sin(c+d*x))^(3/2)/(a+b*cos(c+d*x))^2,x,13,1/2*a*e^(3/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(3/2)*(-a^2+b^2)^(3/4)*d)+1/2*a*e^(3/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(3/2)*(-a^2+b^2)^(3/4)*d)-e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(b^2*d*sqrt(e*sin(c+d*x)))+1/2*a^2*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^2*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))+1/2*a^2*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^2*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))+e*sqrt(e*sin(c+d*x))/(b*d*(a+b*cos(c+d*x)))],
[(e*sin(c+d*x))^(1/2)/(a+b*cos(c+d*x))^2,x,13,-b*(e*sin(c+d*x))^(3/2)/((a^2-b^2)*d*e*(a+b*cos(c+d*x)))+1/2*a*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(e)/((-a^2+b^2)^(5/4)*d*sqrt(b))-1/2*a*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(e)/((-a^2+b^2)^(5/4)*d*sqrt(b))+1/2*a^2*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b*(a^2-b^2)*d*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))+1/2*a^2*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b*(a^2-b^2)*d*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))+sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/((a^2-b^2)*d*sqrt(sin(c+d*x)))],
[1/((a+b*cos(c+d*x))^2*(e*sin(c+d*x))^(1/2)),x,13,-3/2*a*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(b)/((-a^2+b^2)^(7/4)*d*sqrt(e))-3/2*a*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(b)/((-a^2+b^2)^(7/4)*d*sqrt(e))-sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)*d*sqrt(e*sin(c+d*x)))+3/2*a^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))+3/2*a^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))-b*sqrt(e*sin(c+d*x))/((a^2-b^2)*d*e*(a+b*cos(c+d*x)))],
[1/((a+b*cos(c+d*x))^2*(e*sin(c+d*x))^(3/2)),x,14,5/2*a*b^(3/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(9/4)*d*e^(3/2))-5/2*a*b^(3/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(9/4)*d*e^(3/2))-b/((a^2-b^2)*d*e*(a+b*cos(c+d*x))*sqrt(e*sin(c+d*x)))+(5*a*b-(2*a^2+3*b^2)*cos(c+d*x))/((a^2-b^2)^2*d*e*sqrt(e*sin(c+d*x)))-5/2*a^2*b*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^2*d*e*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-5/2*a^2*b*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^2*d*e*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-(2*a^2+3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/((a^2-b^2)^2*d*e^2*sqrt(sin(c+d*x)))],
[1/((a+b*cos(c+d*x))^2*(e*sin(c+d*x))^(5/2)),x,14,-7/2*a*b^(5/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(11/4)*d*e^(5/2))-7/2*a*b^(5/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(11/4)*d*e^(5/2))-b/((a^2-b^2)*d*e*(a+b*cos(c+d*x))*(e*sin(c+d*x))^(3/2))+1/3*(7*a*b-(2*a^2+5*b^2)*cos(c+d*x))/((a^2-b^2)^2*d*e*(e*sin(c+d*x))^(3/2))+1/3*(2*a^2+5*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^2*d*e^2*sqrt(e*sin(c+d*x)))-7/2*a^2*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^2*d*e^2*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))-7/2*a^2*b^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^2*d*e^2*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))],
[1/((a+b*cos(c+d*x))^2*(e*sin(c+d*x))^(7/2)),x,15,9/2*a*b^(7/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(13/4)*d*e^(7/2))-9/2*a*b^(7/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(13/4)*d*e^(7/2))-b/((a^2-b^2)*d*e*(a+b*cos(c+d*x))*(e*sin(c+d*x))^(5/2))+1/5*(9*a*b-(2*a^2+7*b^2)*cos(c+d*x))/((a^2-b^2)^2*d*e*(e*sin(c+d*x))^(5/2))-3/5*(15*a*b^3+(2*a^4-10*a^2*b^2-7*b^4)*cos(c+d*x))/((a^2-b^2)^3*d*e^3*sqrt(e*sin(c+d*x)))+9/2*a^2*b^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^3*d*e^3*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))+9/2*a^2*b^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^3*d*e^3*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-3/5*(2*a^4-10*a^2*b^2-7*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/((a^2-b^2)^3*d*e^4*sqrt(sin(c+d*x)))],
[(e*sin(c+d*x))^(13/2)/(a+b*cos(c+d*x))^3,x,15,11/8*(9*a^4-11*a^2*b^2+2*b^4)*e^(13/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(13/2)*(-a^2+b^2)^(1/4)*d)-11/8*(9*a^4-11*a^2*b^2+2*b^4)*e^(13/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(13/2)*(-a^2+b^2)^(1/4)*d)-11/60*e^5*(5*(9*a^2-2*b^2)-27*a*b*cos(c+d*x))*(e*sin(c+d*x))^(3/2)/(b^5*d)+11/28*e^3*(9*a+2*b*cos(c+d*x))*(e*sin(c+d*x))^(7/2)/(b^3*d*(a+b*cos(c+d*x)))+1/2*e*(e*sin(c+d*x))^(11/2)/(b*d*(a+b*cos(c+d*x))^2)-11/8*a*(9*a^4-11*a^2*b^2+2*b^4)*e^7*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^7*d*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-11/8*a*(9*a^4-11*a^2*b^2+2*b^4)*e^7*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^7*d*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))+11/20*a*(45*a^2-37*b^2)*e^6*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(b^6*d*sqrt(sin(c+d*x)))],
[(e*sin(c+d*x))^(11/2)/(a+b*cos(c+d*x))^3,x,15,-9/8*(7*a^4-9*a^2*b^2+2*b^4)*e^(11/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(11/2)*(-a^2+b^2)^(3/4)*d)-9/8*(7*a^4-9*a^2*b^2+2*b^4)*e^(11/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(11/2)*(-a^2+b^2)^(3/4)*d)+9/20*e^3*(7*a+2*b*cos(c+d*x))*(e*sin(c+d*x))^(5/2)/(b^3*d*(a+b*cos(c+d*x)))+1/2*e*(e*sin(c+d*x))^(9/2)/(b*d*(a+b*cos(c+d*x))^2)+3/4*a*(21*a^2-13*b^2)*e^6*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(b^6*d*sqrt(e*sin(c+d*x)))-9/8*a*(7*a^4-9*a^2*b^2+2*b^4)*e^6*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^6*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))-9/8*a*(7*a^4-9*a^2*b^2+2*b^4)*e^6*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^6*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))-3/4*e^5*(3*(7*a^2-2*b^2)-7*a*b*cos(c+d*x))*sqrt(e*sin(c+d*x))/(b^5*d)],
[(e*sin(c+d*x))^(9/2)/(a+b*cos(c+d*x))^3,x,14,-7/8*(5*a^2-2*b^2)*e^(9/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(9/2)*(-a^2+b^2)^(1/4)*d)+7/8*(5*a^2-2*b^2)*e^(9/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(9/2)*(-a^2+b^2)^(1/4)*d)+7/12*e^3*(5*a+2*b*cos(c+d*x))*(e*sin(c+d*x))^(3/2)/(b^3*d*(a+b*cos(c+d*x)))+1/2*e*(e*sin(c+d*x))^(7/2)/(b*d*(a+b*cos(c+d*x))^2)+7/8*a*(5*a^2-2*b^2)*e^5*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^5*d*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))+7/8*a*(5*a^2-2*b^2)*e^5*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^5*d*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-35/4*a*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(b^4*d*sqrt(sin(c+d*x)))],
[(e*sin(c+d*x))^(7/2)/(a+b*cos(c+d*x))^3,x,14,5/8*(3*a^2-2*b^2)*e^(7/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(7/2)*(-a^2+b^2)^(3/4)*d)+5/8*(3*a^2-2*b^2)*e^(7/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(7/2)*(-a^2+b^2)^(3/4)*d)+1/2*e*(e*sin(c+d*x))^(5/2)/(b*d*(a+b*cos(c+d*x))^2)-15/4*a*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(b^4*d*sqrt(e*sin(c+d*x)))+5/8*a*(3*a^2-2*b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^4*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))+5/8*a*(3*a^2-2*b^2)*e^4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^4*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))+5/4*e^3*(3*a+2*b*cos(c+d*x))*sqrt(e*sin(c+d*x))/(b^3*d*(a+b*cos(c+d*x)))],
[(e*sin(c+d*x))^(5/2)/(a+b*cos(c+d*x))^3,x,14,-3/8*(a^2-2*b^2)*e^(5/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(5/2)*(-a^2+b^2)^(5/4)*d)+3/8*(a^2-2*b^2)*e^(5/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(5/2)*(-a^2+b^2)^(5/4)*d)+1/2*e*(e*sin(c+d*x))^(3/2)/(b*d*(a+b*cos(c+d*x))^2)-3/4*a*e*(e*sin(c+d*x))^(3/2)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x)))-3/8*a*(a^2-2*b^2)*e^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^3*(a^2-b^2)*d*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-3/8*a*(a^2-2*b^2)*e^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^3*(a^2-b^2)*d*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))+3/4*a*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/(b^2*(a^2-b^2)*d*sqrt(sin(c+d*x)))],
[(e*sin(c+d*x))^(3/2)/(a+b*cos(c+d*x))^3,x,14,-1/8*(a^2+2*b^2)*e^(3/2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(3/2)*(-a^2+b^2)^(7/4)*d)-1/8*(a^2+2*b^2)*e^(3/2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(3/2)*(-a^2+b^2)^(7/4)*d)-1/4*a*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/(b^2*(a^2-b^2)*d*sqrt(e*sin(c+d*x)))+1/8*a*(a^2+2*b^2)*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^2*(a^2-b^2)*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))+1/8*a*(a^2+2*b^2)*e^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b^2*(a^2-b^2)*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))+1/2*e*sqrt(e*sin(c+d*x))/(b*d*(a+b*cos(c+d*x))^2)-1/4*a*e*sqrt(e*sin(c+d*x))/(b*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[(e*sin(c+d*x))^(1/2)/(a+b*cos(c+d*x))^3,x,14,-1/2*b*(e*sin(c+d*x))^(3/2)/((a^2-b^2)*d*e*(a+b*cos(c+d*x))^2)-5/4*a*b*(e*sin(c+d*x))^(3/2)/((a^2-b^2)^2*d*e*(a+b*cos(c+d*x)))-1/8*(3*a^2+2*b^2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(e)/((-a^2+b^2)^(9/4)*d*sqrt(b))+1/8*(3*a^2+2*b^2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(e)/((-a^2+b^2)^(9/4)*d*sqrt(b))+1/8*a*(3*a^2+2*b^2)*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b*(a^2-b^2)^2*d*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))+1/8*a*(3*a^2+2*b^2)*e*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/(b*(a^2-b^2)^2*d*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))+5/4*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/((a^2-b^2)^2*d*sqrt(sin(c+d*x)))],
[1/((a+b*cos(c+d*x))^3*(e*sin(c+d*x))^(1/2)),x,14,3/8*(5*a^2+2*b^2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(b)/((-a^2+b^2)^(11/4)*d*sqrt(e))+3/8*(5*a^2+2*b^2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(b)/((-a^2+b^2)^(11/4)*d*sqrt(e))-7/4*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^2*d*sqrt(e*sin(c+d*x)))+3/8*a*(5*a^2+2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^2*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))+3/8*a*(5*a^2+2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^2*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))-1/2*b*sqrt(e*sin(c+d*x))/((a^2-b^2)*d*e*(a+b*cos(c+d*x))^2)-7/4*a*b*sqrt(e*sin(c+d*x))/((a^2-b^2)^2*d*e*(a+b*cos(c+d*x)))],
[1/((a+b*cos(c+d*x))^3*(e*sin(c+d*x))^(3/2)),x,15,-5/8*b^(3/2)*(7*a^2+2*b^2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(13/4)*d*e^(3/2))+5/8*b^(3/2)*(7*a^2+2*b^2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(13/4)*d*e^(3/2))-1/2*b/((a^2-b^2)*d*e*(a+b*cos(c+d*x))^2*sqrt(e*sin(c+d*x)))-9/4*a*b/((a^2-b^2)^2*d*e*(a+b*cos(c+d*x))*sqrt(e*sin(c+d*x)))+1/4*(5*b*(7*a^2+2*b^2)-a*(8*a^2+37*b^2)*cos(c+d*x))/((a^2-b^2)^3*d*e*sqrt(e*sin(c+d*x)))-5/8*a*b*(7*a^2+2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^3*d*e*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-5/8*a*b*(7*a^2+2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^3*d*e*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-1/4*a*(8*a^2+37*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/((a^2-b^2)^3*d*e^2*sqrt(sin(c+d*x)))],
[1/((a+b*cos(c+d*x))^3*(e*sin(c+d*x))^(5/2)),x,15,7/8*b^(5/2)*(9*a^2+2*b^2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(15/4)*d*e^(5/2))+7/8*b^(5/2)*(9*a^2+2*b^2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(15/4)*d*e^(5/2))-1/2*b/((a^2-b^2)*d*e*(a+b*cos(c+d*x))^2*(e*sin(c+d*x))^(3/2))-11/4*a*b/((a^2-b^2)^2*d*e*(a+b*cos(c+d*x))*(e*sin(c+d*x))^(3/2))+1/12*(7*b*(9*a^2+2*b^2)-a*(8*a^2+69*b^2)*cos(c+d*x))/((a^2-b^2)^3*d*e*(e*sin(c+d*x))^(3/2))+1/12*a*(8*a^2+69*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^3*d*e^2*sqrt(e*sin(c+d*x)))-7/8*a*b^2*(9*a^2+2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^3*d*e^2*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))-7/8*a*b^2*(9*a^2+2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^3*d*e^2*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*sin(c+d*x)))],
[1/((a+b*cos(c+d*x))^3*(e*sin(c+d*x))^(7/2)),x,16,-9/8*b^(7/2)*(11*a^2+2*b^2)*arctan(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(17/4)*d*e^(7/2))+9/8*b^(7/2)*(11*a^2+2*b^2)*arctanh(sqrt(b)*sqrt(e*sin(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(17/4)*d*e^(7/2))-1/2*b/((a^2-b^2)*d*e*(a+b*cos(c+d*x))^2*(e*sin(c+d*x))^(5/2))-13/4*a*b/((a^2-b^2)^2*d*e*(a+b*cos(c+d*x))*(e*sin(c+d*x))^(5/2))+1/20*(9*b*(11*a^2+2*b^2)-a*(8*a^2+109*b^2)*cos(c+d*x))/((a^2-b^2)^3*d*e*(e*sin(c+d*x))^(5/2))-3/20*(15*b^3*(11*a^2+2*b^2)+a*(8*a^4-64*a^2*b^2-139*b^4)*cos(c+d*x))/((a^2-b^2)^4*d*e^3*sqrt(e*sin(c+d*x)))+9/8*a*b^3*(11*a^2+2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^4*d*e^3*(b-sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))+9/8*a*b^3*(11*a^2+2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(sin(c+d*x))/((a^2-b^2)^4*d*e^3*(b+sqrt(-a^2+b^2))*sqrt(e*sin(c+d*x)))-3/20*a*(8*a^4-64*a^2*b^2-139*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2))*sqrt(e*sin(c+d*x))/((a^2-b^2)^4*d*e^4*sqrt(sin(c+d*x)))]]:

# Integrands of the form (g Sin[e+f x])^(p/2) (a+b Cos[e+f x])^(m/2)

# Integrands of the form (g Csc[e+f x])^p (a+b Cos[e+f x])^m

# Integrands of the form (g Csc[e+f x])^p (a+b Cos[e+f x])^m when a^2-b^2=0

# Integrands of the form (g Csc[e+f x])^p (a+b Cos[e+f x])^m
